Is it possible to restore a 3rd Party application which has been minimized to the SysTray?
Calling ShowWindow is fine on apps minimized to the TaskBar but where the app has been minimized to the SysTray it appears its handle gets set to zero, and of course ShowWindow can't find it.
No; there are different approaches that can be taken to accomplish this (the application might close the window and open a new copy when restoring, or it might simply hide the original window and show it upon restoring). There is no particular process that an application needs to follow in order to achieve this functionality, so there's no particular process that can be followed in order to reverse it.
If you have information about how a particular application behaves and you'd like to deal with it as a specific case, then that might be possible. But a one-size-fits-all solution unfortunately doesn't exist.
You can try using Process Explorer to determine what (if any) window handles are open by the process. This will at least tell you if it's destroying or hiding the window.